Average Error: 7.2 → 3.2
Time: 20.1s
Precision: 64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.259731899519289 \cdot 10^{-62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\frac{t \cdot z - x}{y}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1.0}\\ \mathbf{elif}\;z \le 1.415907658888434 \cdot 10^{+57}:\\ \;\;\;\;\frac{\left(z \cdot y - x\right) \cdot \frac{1}{t \cdot z - x} + x}{x + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1.0}\\ \end{array}\]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}
↓
\begin{array}{l}
\mathbf{if}\;z \le -7.259731899519289 \cdot 10^{-62}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\frac{t \cdot z - x}{y}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1.0}\\

\mathbf{elif}\;z \le 1.415907658888434 \cdot 10^{+57}:\\
\;\;\;\;\frac{\left(z \cdot y - x\right) \cdot \frac{1}{t \cdot z - x} + x}{x + 1.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{y}{t} + x}{x + 1.0}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r27544970 = x;
        double r27544971 = y;
        double r27544972 = z;
        double r27544973 = r27544971 * r27544972;
        double r27544974 = r27544973 - r27544970;
        double r27544975 = t;
        double r27544976 = r27544975 * r27544972;
        double r27544977 = r27544976 - r27544970;
        double r27544978 = r27544974 / r27544977;
        double r27544979 = r27544970 + r27544978;
        double r27544980 = 1.0;
        double r27544981 = r27544970 + r27544980;
        double r27544982 = r27544979 / r27544981;
        return r27544982;
}
↓
double f(double x, double y, double z, double t) {
        double r27544983 = z;
        double r27544984 = -7.259731899519289e-62;
        bool r27544985 = r27544983 <= r27544984;
        double r27544986 = 1.0;
        double r27544987 = t;
        double r27544988 = r27544987 * r27544983;
        double r27544989 = x;
        double r27544990 = r27544988 - r27544989;
        double r27544991 = y;
        double r27544992 = r27544990 / r27544991;
        double r27544993 = r27544986 / r27544992;
        double r27544994 = fma(r27544993, r27544983, r27544989);
        double r27544995 = r27544989 / r27544990;
        double r27544996 = r27544994 - r27544995;
        double r27544997 = 1.0;
        double r27544998 = r27544989 + r27544997;
        double r27544999 = r27544996 / r27544998;
        double r27545000 = 1.415907658888434e+57;
        bool r27545001 = r27544983 <= r27545000;
        double r27545002 = r27544983 * r27544991;
        double r27545003 = r27545002 - r27544989;
        double r27545004 = r27544986 / r27544990;
        double r27545005 = r27545003 * r27545004;
        double r27545006 = r27545005 + r27544989;
        double r27545007 = r27545006 / r27544998;
        double r27545008 = r27544991 / r27544987;
        double r27545009 = r27545008 + r27544989;
        double r27545010 = r27545009 / r27544998;
        double r27545011 = r27545001 ? r27545007 : r27545010;
        double r27545012 = r27544985 ? r27544999 : r27545011;
        return r27545012;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.2
Target0.4
Herbie3.2
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1.0}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -7.259731899519289e-62

    1. Initial program 12.0

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]
    2. Using strategy rm

    3. Applied div-sub12.0

      \[\leadsto \frac{x + \color{blue}{\left(\frac{y \cdot z}{t \cdot z - x} - \frac{x}{t \cdot z - x}\right)}}{x + 1.0}\]

    4. Applied associate-+r-12.0

      \[\leadsto \frac{\color{blue}{\left(x + \frac{y \cdot z}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}}}{x + 1.0}\]

    5. Simplified4.3

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right)} - \frac{x}{t \cdot z - x}}{x + 1.0}\]
    6. Using strategy rm

    7. Applied clear-num4.5

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{\frac{t \cdot z - x}{y}}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1.0}\]

    if -7.259731899519289e-62 < z < 1.415907658888434e+57

    1. Initial program 0.3

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]
    2. Using strategy rm

    3. Applied div-inv0.4

      \[\leadsto \frac{x + \color{blue}{\left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}}{x + 1.0}\]

    if 1.415907658888434e+57 < z

    1. Initial program 17.8

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]

    2. Taylor expanded around inf 8.2

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1.0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.259731899519289 \cdot 10^{-62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{\frac{t \cdot z - x}{y}}, z, x\right) - \frac{x}{t \cdot z - x}}{x + 1.0}\\ \mathbf{elif}\;z \le 1.415907658888434 \cdot 10^{+57}:\\ \;\;\;\;\frac{\left(z \cdot y - x\right) \cdot \frac{1}{t \cdot z - x} + x}{x + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t} + x}{x + 1.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))